Constraint Programming models for the parallel drone scheduling vehicle routing problem

Question

1. What are the extensions of the original Parallel Drone Scheduling Vehicle Routing Problem (PDSTSP) that involve multiple trucks and more realistic constraints?

2. What factors limit drone service in MT-PDSVRP?

3. What variables are included in MT-3IDX model?

4. What is the purpose of the 'MultipleCircuit' statement in MT-2IDX model?

Accepted Answer

Extensions of the original PDSTSP that involve multiple trucks and more realistic constraints include the Min-Time Parallel Drone Scheduling Vehicle Routing Problem (MT-PDSVRP) and the Min-Cost Parallel Drone Scheduling Vehicle Routing Problem (MC-PDSVRP). The MT-PDSVRP is a straightforward extension where multiple trucks are employed, and the target is to minimize the time required to complete the delivery to the last customer serviced and go back to the depot. The MC-PDSVRP, on the other hand, introduces concepts such as capacity, load balancing, and decoupling of costs and times. Both problems have been explored using Constraint Programming models, with experimental results showing the potential of solvers in dealing with TSP and VRP problems.

Accepted Answer

In the Min-Time Parallel Drone Scheduling Vehicle Routing Problem (MT-PDSVRP), drone service is limited by factors such as parcel weight, distance from the depot, and terrain obstacles. Not all customers can be served by drones due to these limitations. The set of customers that can be served by drones, denoted as C D C, are referred to as drone-eligible. These customers are those that meet the criteria for drone service, considering the weight of the parcel, distance from the depot, and terrain obstacles like hills or high-rise buildings. The objective of the MT-PDSVRP is to minimize the time required for all deliveries and ensure all trucks and drones return to the depot. The solution involves trucks and drones working in parallel, with the objective function focusing on minimizing the time required by the vehicle with the longest total operational time. An example solution is provided in Figure 1, showing the tours of trucks and missions of drones in a small instance.

Accepted Answer

MT-3IDX model includes binary variables z kij, x di, and continuous variable a. z kij represents if node i is visited before node j in truck k's tour, x di indicates if customer i is visited by drone d, and a is a continuous variable for the objective function. These variables are used to form a valid tour and minimize the time taken by the latest vehicle to complete tasks.

Accepted Answer

The 'MultipleCircuit' statement in MT-2IDX model ensures that the y variables take values in a way to form valid truck tours and self-loops for variables associated with customers visited by drones. It is used to impose constraints on the y variables, ensuring that each customer is visited either by a truck or a drone. This statement plays a crucial role in maintaining the validity of the truck tours and drone operations within the model.

Accepted Answer

In the Min-Cost Parallel Drone Scheduling Vehicle Routing Problem (MC-PDSVRP), several elements are added to the MT-PDSVRP model. These include transportation costs for trucks and drones, parcel weights, capacity constraints for trucks, and maximum working time limits for trucks and drones. These elements contribute to a more complex objective function and additional constraints in the problem. The transportation costs, denoted as c T ij for trucks and c D i for drones, are used to define a new objective function to be minimized. Each customer's delivery request is associated with a parcel weight w i. Truck capacity constraints limit the transportation of at most Q T units of weight in its route. Additionally, there are upper bounds t T and t D on the maximum working time for trucks and drones, respectively. These constraints are typically used to introduce load-balancing concepts into the optimization process, ensuring efficient and balanced scheduling of drone and truck operations.

Accepted Answer

The objective function (20) aims to minimize the total cost incurred to service all customers. It considers the costs associated with each truck and drone used in the service. By minimizing this cost, the model seeks to optimize the allocation of resources and ensure efficient service delivery to customers. The objective function takes into account various factors such as the number of trucks and drones, their working times, and capacity constraints. Overall, it plays a crucial role in determining the most cost-effective solution for the given problem.

Accepted Answer

The variables used in the MC-2IDX model are the same as in the MT-2IDX model, with the addition of new continuous variables b i and i V. These variables count the weight carried by each truck k T. The objective function minimizes the total cost, considering the costs of each truck and drone. Constraints ensure valid truck tours, customer visits by trucks or drones, limited truck tours, drone operating time, weight and time limits, and variable domain definitions.

Accepted Answer

The experiments were run on a laptop computer equipped with 32 GB of RAM and an Intel Core i7 12700F CPU with 12 cores (8 with a maximum frequency of 4.9 GHz and 4 with a maximum frequency of 3.6 GHz).

Accepted Answer

The results of MT-PDSVRP benchmarks, obtained using a branch and cut (BC) approach and nine variations of a hybrid metaheuristic approach (HM), indicate that the new CP-based approaches are competitive with state-of-the-art results. The BC solver was run on a computer with an Intel Xeon CPU E5-2670 CPU with 2x8 cores running at 2.6 GHz and 62.5 GB of RAM, with a maximum computation time of 10800 seconds. The HM variations were run on a computer with an Intel core i5-6200U CPU running at 2.4 GHz and 8 GB of RAM, with a maximum computation time of 1000 for each variation. The results showed that new improved lower bounds were provided for all the instances considered, and two new best-known heuristic solutions were retrieved. The comparison between the CP models revealed that the model MT-3IDX, characterized by the set of variables z with 3 indices, performed better, providing substantially tighter lower bounds. However, it was not capable of handling the last instance, which required a larger number of z variables. In this case, the model MT-2IDX, with its smaller memory footprint, was the only viable option. The quality of the heuristic solutions produced by the CP methods was consistently high, although not always matching the state-of-the-art provided by the nine purely heuristic methods summarized in column HM.

Accepted Answer

The instances in Raj et al. [16] were originated from TSPLIB [17] and defined according to previous work in [6] for the PDSTSP. Instances with more than one truck were reported, with customer numbers ranging between 48 and 229. Full details can be found in [16].

Accepted Answer

The instances in MC-PDSVRP are based on statistical data shared by logistics service providers and common practices. They are generated as random but can be considered realistic. The number of customers considered are 30, 50, 100, 200, and 400. The details of these instances, including the number of trucks, drones, loading capacities, speeds, and battery endurance, are provided in [11]. The results of experiments are summarized in Tables 4, 5, 6, 7, and 8, divided according to the size of the instances.

Constraint Programming models for the parallel drone scheduling vehicle routing problem

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Most frequently asked questions

1. What are the extensions of the original Parallel Drone Scheduling Vehicle Routing Problem (PDSTSP) that involve multiple trucks and more realistic constraints?

2. What factors limit drone service in MT-PDSVRP?

3. What variables are included in MT-3IDX model?

4. What is the purpose of the 'MultipleCircuit' statement in MT-2IDX model?

5. What elements are added to the MT-PDSVRP?

6. What is the objective function in M C−3IDX : min?

7. What variables are used in MC-2IDX model?

8. What hardware specifications were used for computational experiments?

9. What are the results of MT-PDSVRP benchmarks?

10. What instances were used in Raj et al. [16]?

11. What are the instances considered in MC-PDSVRP?

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